Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{z^2 - 36}{z + 6}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = z$ $ b = \sqrt{36} = 6$ So we can rewrite the expression as: $t = \dfrac{({z} + {6})({z} {-6})} {z + 6} $ We can divide the numerator and denominator by $(z + 6)$ on condition that $z \neq -6$ Therefore $t = z - 6; z \neq -6$